Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

An activity making various patterns with 2 x 1 rectangular tiles.

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

This activity investigates how you might make squares and pentominoes from Polydron.

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

Can you use the information to find out which cards I have used?

What is the best way to shunt these carriages so that each train can continue its journey?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

How many models can you find which obey these rules?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

Can you find all the different ways of lining up these Cuisenaire rods?

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

What happens when you try and fit the triomino pieces into these two grids?

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

How many trains can you make which are the same length as Matt's, using rods that are identical?

Can you find out in which order the children are standing in this line?

What could the half time scores have been in these Olympic hockey matches?

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

Investigate the different ways you could split up these rooms so that you have double the number.

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?